Passive scalars, moving boundaries, and Newton's law of cooling

TL;DR

This paper investigates how passive scalars evolve within moving and rigid domains under Newton's law of cooling, analyzing equilibrium states and how boundary conditions influence the scalar's relaxation rate.

Contribution

It introduces a comprehensive analysis of passive scalar dynamics with Robin boundary conditions in moving domains, extending understanding of scalar equilibration in complex geometries.

Findings

Equilibrium configurations depend on boundary parameters and domain motion.

The scalar's relaxation rate is influenced by boundary conditions and background flow.

Moving boundaries significantly affect scalar equilibration dynamics.

Abstract

We study the evolution of passive scalars in both rigid and moving slab-like domains, in both horizontally periodic and infinite contexts. The scalar is required to satisfy Robin-type boundary conditions corresponding to Newton's law of cooling, which lead to nontrivial equilibrium configurations. We study the equilibration rate of the passive scalar in terms of the parameters in the boundary condition and the equilibration rates of the background velocity field and moving domain.